Although in metrizable spaces all types of completeness that we mentioned above are the same, in general non-metrizable spaces the picture is much more complex. It is well-known that an incomplete topological vector space may be sequentialy complete. The most important example of such kind is the Hilbert space \(\ell _2\)  equipped with the weak topology. This section is devoted to the non-metrizable case, where many interesting examples come from the duality theory for locally convex spaces. A very good comprehensive introduction to duality is [1]}, a shorter one may be found in [2]}. As usual, for a duality pair \(X, Y\)  we denote \(\sigma (X, Y)\)  the weak topology on \(X\)  generated by \(Y\) .
